Suppose $A \in B(H)$ is a self-adjoint bounded operator on some Hilbert space $H$ and $V \leq H$ is some finite-dimensional subspace. Let $P_V$ be the orthogonal projection from $H$ to $V$. I'm interested in the following: is the spectrum of a matrix $P_V \circ A\big|_{V}$ contained inside the spectrum of the operator $A$?
I'm looking into this because it would help me solve the problem I'm working on (for context, see my post history).
Edit: forgot to add that $A$ is self-adjoint.
No. Let $H=\Bbb{R}^2$, $V=\operatorname{span}\{(1,0)\}$, $$A = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$$ Then $P_V \circ A\big|_{V} = 0$ so $\sigma(P_V \circ A\big|_{V})= \{0\}$ but $\sigma(A)=\{\pm 1\}$.